As the comments have noted, this is a rather unclear question. However, there's an important sense in which the answer is yes.
I'll write "N" for the set of natural numbers and N for the structure of the natural numbers with zero, successor, addition, and multiplication.
Specifically, looking at a function F from N to N for simplicity, the following are equivalent:
F is computable by a Turing machine (or equivalently, per the comments above, any Turing-complete programming language - which is all of the ones you'd actually use).
F is representable in first-order Peano arithmetic PA (or indeed even less): there is a formula p(x,y) such that for each natural number n, there is exactly one natural number m such that PA proves p(n,m), and moreover that natural number is exactly f(n). Here for simplicity I'm conflating a number with the corresponding numeral - e.g. 2 versus S(S(0)).
F is Sigma^0_1-definable in N.
(Similar results hold for higher-arity functions, partial functions, and relations.)
So we have both provability and definability criteria, tied to first-order theories and structures respectively, which exactly capture computability. And indeed this is only a tiny fragment of first-order logic: there are non-computable sets of natural numbers (e.g. the Halting Problem) which are nonetheless first-order definable in N. Really, I would say that models of computation correspond to a very tiny fragment of first-order logic specifically in the context of N or similar structures.
That said, this isn't really the whole picture. Logics stronger than first-order logic are still "computable on finite structures" - e.g. we can brute-force determine whether a second-order sentence with k second-order quantifiers holds in a structure of size n in time more-or-less 2^(kn) - and this plays an important role in descriptive complexity theory - the point being that we can often view individual inputs to algorithms as finite structures. By contrast, the above section is looking at how the whole "input space" is construed.