When D is an integral domain with field of fractions K , the ring Int ( D ) = { f ( x ) ∈ K [ x ] | f ( D ) ⊆ D } of integer-valued polynomials over D has been extensively studied. We will extend the integer-valued polynomial construction to certain non-commutative rings. Specifically, let i , j , and k be the standard quaternion units satisfying the relations i 2 = j 2 = − 1 and i j = k = − j i , and define Z Q : = { a + b i + c j + d k | a , b , c , d ∈ Z } . Then, Z Q is a non-commutative ring that lives inside the division ring Q Q : = { a + b i + c j + d k | a , b , c , d ∈ Q } . For any ring R such that Z Q ⊆ R ⊆ Q Q , we define the set of integer-valued polynomials over R to be Int ( R ) : = { f ( x ) ∈ Q Q [ x ] | f ( R ) ⊆ R } . We will demonstrate that Int ( R ) is a ring, discuss how to generate some elements of Int ( Z Q ) , prove that Int ( Z Q ) is non-Noetherian, and describe some of the prime ideals of Int ( Z Q ) .