In this paper, we present a candidate of a vector space basis for the noncommutative algebra O(S _q ^4n-1) of the quantum symplectic sphere for every n ≥ 1. The algebra (S _q ^4n-1) is defined as a certain subalgebra of the quantum symplectic group O(SP _q(2n)). A nontrivial application of the Diamond Lemma is used to construct the vector space basis and the conjecture is supported by computer experiments for n = 1, 2,⋯, 8.