In this paper, we utilize the perspective of property testing to consider the testability of relational database queries. A primary motivation is the desire to avoid reading an entire database to decide a property thereof. We focus on conjunctive queries, which are the most basic and heavily studied database queries. Each conjunctive query can be represented as a relational structure A such that deciding if the conjunctive query is satisfied by a relational structure B is equivalent to deciding if there exists a homomorphism from A to B. We phrase our results in terms of homomorphisms. Precisely, we study, for each relational structure A, the testability of homomorphism inadmissibility from A. We consider algorithms that have oracle access to an input relational structure B and that distinguish, with high probability, the case where there is no homomorphism from A to B, from the case where one needs to remove a constant fraction of tuples from B in order to suppress all such homomorphisms. We provide a complete characterization of the structures A from which one can test homomorphism inadmissibility with one-sided error by making a constant number of queries to B. Our characterization shows that homomorphism inadmissibility from A is constant-query testable with one-sided error if and only if the core of A is alpha-acyclic. We also show that the injective version of the problem is constant-query testable with one-sided error if A is alpha-acyclic; this result generalizes existing results for testing subgraph-freeness in the general graph model.