We study the existence of positive functions K in C^1(S^n) such that the conformal Q-curvature equation Pm(v) = K v^{n*} on S^n has a singular positive solution v whose singular set is a single point, where m is an integer satisfying 1 <= m < n/2 and Pm is the intertwining operator of order 2m. More specifically, we show that when n => 2 m + 4, every positive function in C^1(S^n) can be approximated in the C^1(S^n) norm by a positive function K in C^1(S^n) such that the equation has a singular positive solution whose singular set is a single point. Moreover, such a solution can be constructed to be arbitrarily large near its singularity.