There is a difference. Complementarity concerns a pair of observables in quantum physics that are represented in Heisenberg's uncertainty relation, e.g. position and momentum, or energy and time. Conjugacy refers to the technical representation of these observables. In Lagrangian field theory, a pair of complementary observables is represented by conjugate field operators, and this is what links the two expressions to each other (take the Lagrangian of a free non-relativistic particle and check for the conjugacy of position and momentum, as a simple example). Briefly, one is more phenomenological, the other is more technical. The colleague you mention is right in the sense that the two expressions are often used synonymously, but in a stricter reading the difference is as I stated above.

To add to Timo, conjugate momentum is a term coming from Hamiltonian mechanics. On the phase spaces of classical Hamiltonian mechanics (depending on your sophistication $R^{2n}$, the cotangent space $T^*M$ of a manifold, or a general symplectic manifold) we can always choose coordinates q_1, \ldots q_n, p_1, \ldots p_n, such that the Poisson brackets are $\{q_i, q_j\} = 0$, $\{p_i, p_j\} = 0$, and $\{q_i, p_j\} = \delta_{ij}$. The p_i is then called the conjugate momentum of the coordinate q_i (same index i). Example spinning wheel: q = \theta= angle, p = J= angular momentum. That such coordinates always exist locally is the content of the Darboux theorem in symplectic geometry). By the correspondence principle, after quantization (which is not a totally well defined operation) we have operators q_i, p_j satisfying the canonical commutation relations $[q_i, q_j] =0$, $[p_i, p_j] = 0$, $[p_i, q_j] = i\hbar$. The operator $p_i$ is still called the conjugate momentum of $q_i$. Moreover one can show (exactly as for the classical uncertainty relation) that the non commutation of the conjugate coordinate and momentum operator implies a Heisenberg uncertainty type relation $ \ge (1/2) \hbar$. See the section Robertson - Schrödinger uncertainty relation in the Wikipedia article https://en.wikipedia.org/wiki/Uncertainty_principle

I just noticed that this question has already been discussed in the three https://www.researchgate.net/post/Which_physical_variables_do_not_exist_in_conjugate_pairs_and_do_not_have_an_uncertainty_relationship_in_quantum_mechanics_Can_you_list_which_pairs_exist_and_list_those_variables_that_dont_exist_in_pa?exp_tc=tprc

We say that two observables are complementary if theirs corresponding quantum observables have no common eigenvectors. If you want more about complementary observables you can look at the page 27 in the book of Beltrametti and Cassinelli "The Logic of Quantum Mechanics"