No.
Monitor refresh is only slightly related to frame rendering rate.
The early stage of the graphics card actually renders the frames. The speed this occurs depends on how heavy the graphics are and how strong your GPU is.
The second stage actually outputs it to the monitor and this will be a fixed rate usually in the 60-120hz range depending on the monitor.
On strong GPUs or non-demanding games, the rendering stage of the GPU may render frames faster then they can be outputed, in which case some of the frames are discarded.
Conversely, if your GPU is working hard, it will not be able to render as fast as the monitor refresh, but the last frame will be kept on screen until the rendering is finished.
Framerate can be much higher then monitor refresh but there is no benefit to it.
If VSYNC is enabled, the rendering part will deliberately throttled back to lock the framerate to the monitor refresh rate.