Can silicon dioxide melt at 20 °C?
Question: Can $\ce{SiO2}$ melt at $\pu{20 ^\circ C}$?

According to experimental and calculated data values, my answer is no. See the phase diagram of pure silica based on the experimental and calculated data given in Ref.1:

Reference 1 states that:

An internally consistent data set on the thermodynamic properties of the silica polymorphs stable up to $\pu{15 GPa}$ (α‐quartz, β‐quartz, tridymite, cristobalite, coesite, and stishovite) and the liquid phase is presented. The data set was produced through a computer‐based assessment of the properties in which the available thermochemical (calorimetric), physical (bulk modulus and thermal expansion), and solid‐state and melting transition data (including some newly determined data on the high‐pressure polymorphs cosite and stishovite) were considered. The data set can be used to calculate phase relations at pressures of $\pu{0.1 MPa}$ to $\pu{15 GPa}$ and temperatures of $300$ to $\pu{3200 K}$. The calculated phase diagram using these data agrees quite well with the phase equilibrium determinations except for the high‐temperature part of the coesitestishovite boundary. The properties of the liquid phase obtained are also in good agreement with the available data.

Keep in mind that $\pu{0.1 MPa} \approx \pu{1 atm}$. According to the phase diagram, cristobalite form is in liquid form at temperature grater than $\pu{1750 K}$ in pressure range of $\pu{0.1-0.7 MPa}$. The extrapolation of that boundary would show it would never cross $\pu{20 ^\circ C}$ point on $x$-axis without going to negative pressure. Therefore, it is impossible to have any form of $\ce{SiO2}$ in reduced or high pressusituations at room temperature or below.
References:

V. Swamy Surendra, K. Saxena, Bo Sundman, J. Zhang, “A thermodynamic assessment of silica phase diagram,” Journal of Geophysical Research 1994, 99(B6), 11787-11794 (https://doi.org/10.1029/93JB02968).