Let \(G\) be a compact group, and let \(H\) be its closed subgroup such that the quotient \(G/H\) is infinite. An \(H\)-biinvariant continuous function \(f:G\to\mathbb{C}\), the space of all such functions being denoted by \(C(H\smallsetminus G/H)\), is said to be strictly positive definite if and only if \(\sum^n_{i,j=1}c_i\overline{c_j}f(x_i\overline{x_j})>0\) for any finite set \(\{x_1,x_2,\dots,x_n\}\subseteq G\) such that the cosets \(x_1H,x_2H,\dots,x_nH\) are distinct, and any \(c_1,c_2,\dots,c_n\in\mathbb{C}\) not all equal to zero. For any irreducible unitary representation \(\pi\) of \(G\), we denote its space by \({\mathcal H}_\pi\) and the subspace of \(H\)-fixed vectors by \({\mathcal H}^H_\pi\). Finally, let \((G/\widehat H)=\{\pi\in\widehat G:{\mathcal H}^H_\pi\neq 0\}\). The authors prove: a function \(f\in C(H\smallsetminus G/H)\) is strictly positive definite if \(\pi(f)\geq 0\) for all \(\pi\in(G/\widehat H)\), and \(\pi(f)|_{{\mathcal H}^H_\pi}>0\) for all but finitely many \(\pi\in(G/\widehat H)\). When \((G,H)\) is a Gelfand pair, the operator \(\pi(f)\) is a constant, denoted by \(\lambda_\pi(f)\), a multiple of the projection onto the subspace \({\mathcal H}^H_\pi\), and then a function \(f\in C(H\smallsetminus G/H)\) is strictly positive definite if \(\lambda_\pi(f)>0\) for all but finitely many \(\pi\in(G/\widehat H)\). This generalizes a result of \textit{M. Schreiner} [Proc. Am. Math. Soc. 125, 531-539 (1997; Zbl 0863.43002)], concerning strictly positive definite functions on a sphere in a Euclidean space.