For a harmonic oscillator, the graph between momentum p and displacement q would come out as

For a simple harmonic oscillator (SHO), the relationship between momentum (p) and displacement (q) is governed by the conservation of total energy. The total energy E is the sum of kinetic energy (p"/2m) and potential energy (kq"/2). This results in the equation E = p"/2m + kq"/2, which is mathematically equivalent to the standard equation of an ellipse. In phase space, where momentum is plotted on the vertical axis and displacement on the horizontal axis, the trajectory of an undamped oscillator follows a closed elliptical path. For small oscillations, even a pendulum's motion appears as an ellipse in phase space [1]. If the angular frequency and mass are scaled such that the coefficients of p and q are equal, the ellipse may become a circle, but in the general physical case of p versus q, the trajectory is defined as an ellipse.

Sources

1. [1] https://www.damtp.cam.ac.uk/user/tong/dynamics/four.pdf