The rocky mountains run right through the state, and so it is difficult to count. It is generally accepted that there are 53 peaks above 14,000 feet. As for the rest of the mountains, it depends on how you qualify what a mountain is... how high one peak is saddle point from the next peak to qualify as 'a mountain' (called 'Prominence').
The link I've provided below identifies and ranks the 100 mountains in Colorado with the largest primary factor or (prominence) of a mountain in USA. Use this link for more information about the ranking process and their names.
There are FAR more than 100 mountains in Colorado. If you count just the high points above 13000 feet there are over 700! Then there is the myriad of smaller peaks. Of course the prominence of these points is not always significant, but it is generally accepted that the minimum acceptable prominence is ~100 m.