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The existence of a partial quadrangle PQ(s, t, μ) is equivalent to the existence of a diamond-free strongly regular graph SRG(1 + s(t + 1)+s²t(t + 1)/μ, s(t + 1), s - 1, μ). Let S be a PQ(3, (n + 3)(n² - 1)/3, n² + n) such that for every two noncollinear points p1 and p2, there is a point q noncollinear with p1, p2, and all points collinear with both p1 and p2. In this article, we establish that S exists only for n ∈ {-2, 2, 3} and probably n = 10. [ABSTRACT FROM AUTHOR]