Hello
I believe this may be what you looking for?
Functions to be used later
exampleF1[u_, v_] := u Cos[v]
exampleF2[u_, v_] := ArcTan[u, v]
exampleG1[u_, v_] := (u + v) u
exampleG2[u_, v_] := (u - v) Exp[v]
Function to create your equation, and example usage
myEquation[f_, g_, q_, t_] := f[q, t] == g[1/t, 1/q]
myEquation[exampleF1, exampleG1, q, t]
myEquation[exampleF2, exampleG1, q, t]