DSolve can solve if : z1 = z2 = z
Executed on Mathematica 13.1:
DSolve[{v1''''[z] + f v1''[z] + k v1[z] - q == 0,
v2''''[z] + f v2''[z] == q, v1[0] == q/k, v1'[0] == 0,
v1[a1] == v2[0], v1'[a] == v2'[0], v1''[a] == v2''[0],
v1'''[a] == v2'''[0], v2''[a2] == 0}, {v1[z], v2[z]}, z]
(*A very very long answer.....See attached file.*)
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