That's not even close to true. And what makes you say that there is no sun during monsoon season? That's traditionally when the sun is at its peak for the area.
But people do certainly get colds in the summer. Many do. And while there's the tiniest bit of merit to your claims, it's only in that people with low vitamin D are more susceptible.
But yes, people can and do get colds at any time of the year. It is a virus, and infection with the virus will (usually) result in symptoms. There certainly is such a thing as flu season, and most of it actually is, in a way, because aren't getting much sun. But more accurately, it's because people are staying indoors more - the drier air and the increase in the amount of time that people spend in close proximity together are perfect circumstances for this virus to thrive and spread.
The drop in vitamin D MIGHT make people more susceptible to it, but even then, it's a relatively small effect compared to the above. And just about anyone living at a latitude that has snow for even a month most years should be supplementing anyways. However, good humidifiers and proper sanitation would have a far greater impact. Unfortunately, the former is really only in your control at home (and maybe at work), and with the biggest factor of all - proper sanitary practices - you only have control over yourself, which can be rendered moot by a family member or coworker with more lax standards.