Using the word list provided by Golden Dragon here are the frequencies for every pair of characters.
The word list had 172820 words including plurals.
There were 84 pairs that weren't found in any word.
cg,qm,jr,jq,zr,kq,gx,tx,ql,mx,jp,kx,gq,zf,kz,vb,qt,qk,jm,jl,wv,vc,jt,jh,vf,vh,jg,jf,qv,bq,yq,jc,jb,qj,qx,bx,vj,fz,fx,fv,vm,qy,cf,qn,qz,xz,cj,rx,xx,fq,jv,vp,vq,wx,jw,xr,cv,cx,vt,qh,qg,jx,jy,zx,vw,vx,jz,xk,pq,qf,xj,pv,px,qr,qp,sx,qq,pz,hx,dx,qd,qc,wq,qb
Below are plots of the pair frequencies. A is top left, Z is bottom right.
Most common is white:

Those that occur more than 512 times or more are pure white in this one:

Because of what I had hoped to do with this data regarding word searches, think the pairs need to be treated as reversible, and to list of pairs that occur 0 times needs to extended to include runs of 3 and 4 letters.
Here is the data in machine and human readable formats:
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0 - zf kq jl zx qp fv jz qj kx qy jh xx jp vt vh cv qn vq xz wx vc jt xk qt jm bq jb tx wq qz zr hx vm jw jg vb pq qm sx gq jv qr qf ql gx qc qx cj mx jr qd qg px jf qv vw qk fz bx qq qh cf jx xr cg pv vj vx cx xj vp jy fx qb wv yq fq kz jc jq rx dx pz vf
1 - qe jk fk
2 - xv zt jd zd qw js yy wj xd hv cw zg vd qo vz gv cp zj
3 - bz vk
4 - zk mq zb gj jj cb fg fj xg jn zc lj zq zs gz xn
5 - qs xq sz lx vg
6 - vn gk hz tq mz mj mg hq
7 - qa zn zv
8 - fw vl xm dq zh hj uw
9 - zw wz qi
10 - zp fc mk zm yj cm
11 - kj dk
12 - lq fm bk xb fd pj
13 - fp
14 - md gc uq
15 - xw
16 - lz
17 - wg vs
18 - yv xf
20 - kv kg
21 - cz
22 - tk
23 - hg fh
24 - fn gp
26 - vv fb ww
27 - dz wc bg
28 - tj iy pk hk
29 - yk uh
30 - uu cd xl mt
32 - mh
33 - uj
34 - tv vr
36 - mv kc
38 - kd mw
41 - pg xs
42 - wu bw
43 - pd
45 - hh
46 - hc mc
47 - ij nx kk sj
48 - sv
49 - bf
50 - gd zu
51 - cn
52 - bp
53 - uy
54 - rz
55 - ii hd mr
57 - bh
58 - vy iw yx
59 - wp gf
65 - pc
66 - kp dt rj
67 - wm bv rq
68 - wy
69 - bn
71 - lr
72 - pf
74 - yu wf
75 - gw yh
76 - aa yf
77 - pw
78 - hp
79 - wt
80 - gt
81 - oj
88 - pm cq
89 - bm
93 - pb
96 - wk
97 - kf
100 - xh gb kt
101 - km hf
103 - ao
106 - mf
107 - yz
113 - lw dc
119 - sg
121 - dp dj
122 - aq ih
124 - aj
125 - bj
129 - oq
130 - bc
131 - kw
137 - sd lh
139 - bd pn
145 - uz
146 - yw
149 - td
150 - ku
152 - ux
154 - wd wb tg
156 - df
158 - kr
160 - xu
161 - kb
162 - hw hb
165 - zl
170 - kh
171 - nz nq
172 - ej
174 - ml sr
182 - zy
183 - dv
190 - uv
196 - dh
206 - ez
207 - ji
209 - tp
210 - nj
213 - bt
215 - uk
223 - vu
224 - tz
227 - db
230 - yb
239 - xy
242 - iq by
244 - yg
253 - sb
262 - dw
264 - sf
269 - lg
273 - lb
275 - dm
278 - xc xo
308 - oh
310 - fs
314 - gm
316 - ek
317 - mn
323 - tb
324 - nw
331 - oz
332 - ky
339 - ix
346 - wl zz
349 - xa
353 - ko
354 - tn
355 - rw
360 - hm
369 - tm
387 - tf
397 - wr
403 - kn
409 - sq
424 - fy
426 - eq
428 - lp
429 - lv
436 - uo
438 - np
441 - nm
451 - nb
452 - lf
467 - hn
471 - yd
482 - nr
483 - nh
488 - lc
508 - jo
511 - ah
516 - uf
518 - lk
526 - ln
527 - lm
528 - je
537 - oy
539 - xp
543 - tw
549 - ax
561 - py
562 - ws
563 - xt
574 - dg
584 - rh
587 - ny
589 - ja
593 - nl
602 - yo
608 - kl wn
619 - dy
625 - zo
626 - yr
627 - dn
628 - eh
629 - hl
645 - ju
654 - gy
672 - ft
679 - iu
686 - xe
701 - ya
705 - az
709 - rf
724 - yn
737 - my
758 - wh
760 - sw
769 - rv
771 - cs
773 - yi
780 - nv
785 - hs
806 - yt
841 - ox
863 - yc
867 - ik ye
876 - bs
887 - ka
891 - ey
903 - ae
905 - ym
917 - xi
940 - ok
946 - sk
954 - bb
959 - of
963 - cc
968 - dd
1001 - af
1011 - tc
1071 - aw
1113 - yl
1117 - rk
1123 - ld
1127 - cy
1140 - oe
1141 - ht
1152 - hu
1165 - gn
1178 - vo
1179 - rl
1195 - ug
1203 - nk
1221 - sy
1249 - gg
1257 - ew
1265 - eu
1270 - ak
1275 - eb
1291 - yp
1319 - nf
1329 - zi
1343 - za
1347 - fr
1372 - ay
1393 - hr dl
1408 - rp
1411 - ud
1439 - nu
1463 - rb
1498 - rg
1516 - ys
1560 - gh
1612 - ks
1630 - tl
1633 - sn
1634 - du
1637 - wo
1651 - ue
1652 - av
1656 - ev
1660 - gu
1661 - ff
1694 - pt
1695 - nn pu
1723 - oa
1763 - sl
1765 - eo
1766 - ei
1776 - gs
1784 - wi
1793 - mm
1841 - ui
1843 - fa
1857 - we
1862 - ob ua
1870 - pp
1876 - lt
1880 - rn
1903 - go
1934 - fu
1943 - mu
1948 - ps
1959 - ef
1985 - ib
2026 - eg
2029 - dr
2042 - mb
2090 - ub
2135 - uc
2139 - fl
2187 - au
2195 - bu
2234 - oi
2244 - rc
2283 - gl
2294 - up
2343 - va
2387 - hy
2392 - fo
2421 - ms
2439 - cl
2449 - ry wa
2455 - fe
2474 - rr
2483 - ex
2506 - qu
2511 - ow
2526 - ki
2528 - br
2533 - if
2544 - rd
2613 - ty
2629 - ip
2633 - rm
2707 - ls
2718 - ds
2722 - da
2866 - ru
2892 - ov
2903 - od
3049 - sm
3053 - tt
3062 - lu
3168 - do
3194 - tu
3203 - ga
3243 - ze
3246 - cu
3250 - ai
3297 - ig
3312 - vi
3350 - ir
3368 - mp
3397 - pl
3542 - ep
3558 - bo
3572 - um
3601 - gr
3675 - fi
3710 - ag
3760 - sa
3762 - so rt
3773 - cr
3815 - iv
3822 - oo
3829 - sp
3842 - oc
3861 - ba
3866 - ke
3878 - ee
3904 - ck
3942 - gi
3972 - be
3978 - sc
4078 - im
4212 - su
4272 - bi
4367 - ap
4454 - ad
4522 - iz
4523 - pi
4668 - og
4694 - ct
4829 - am
4897 - mo
4905 - ut
4920 - id
4938 - op
4966 - pa
4971 - em
4999 - ph
5060 - bl
5174 - ot
5254 - ci
5259 - po
5413 - ab
5442 - ha
5513 - sh
5616 - ge
5757 - hi
5772 - ul
5781 - nc
5893 - ia
5909 - pr
6010 - no
6031 - th
6045 - ho
6078 - os
6089 - ce
6130 - ec
6194 - ur
6288 - ts
6346 - nd
6400 - om
6578 - us
6754 - ea
6798 - ac
7042 - na
7411 - ma
7574 - si
7654 - il
7707 - ly
7710 - pe
7789 - et
7805 - un
7821 - ch
7906 - ol
7919 - lo
7986 - ou
8017 - el
8085 - mi
8129 - ve
8215 - ni
8258 - to
8279 - he
8313 - ll
8333 - me
8437 - as
8508 - di
8555 - tr
8619 - ta
9217 - ca
9860 - rs
10163 - ns
10367 - ie
10372 - la
10554 - de
10626 - io
10884 - co
11034 - ss
11172 - it
11703 - ro
12409 - or
12621 - nt
12946 - ic
13056 - ar
13111 - se
14857 - an
14947 - ra
15027 - ri
15180 - ne
15239 - li
15383 - le
15552 - is
15761 - al
16378 - st
16978 - en
17018 - ed
17683 - ng
19348 - at
19890 - on
19901 - re
20028 - te
24411 - ti
32289 - er
32894 - in
38404 - es