$λ$-quiddities of size $n$ are $n$-tuples of elements from a fixed set that are solutions to a matrix equation which is fundamental in the study of the combinatorics of the modular group and Coxeter's friezes. To gain further insight into these objects, we use a notion of irreducibility, which allows restricting the study to a limited number of elements that must be determined for each set. Our goal here is to define several families of $λ$-quiddities over finite fields and to study their irreducibility properties, with the specific aim of establishing lower bounds on the maximal size of irreducible elements over $\mathbb{F}_{q}$.