Converting from Pantone Color to RGB or Hex

I'm trying to resolve a color variance. When we develop web pages in our office that are to match printed brochures, colors within tables and text colors are different between our machines. For example, a red similar to a PMS 186 on one screen is pretty pink on mine. A reflex blue tone on my screen is darker than on a colleague's. - RN

Since PMS (Pantone® Matching System) colors are exclusive to printing inks, the Web-based equivalents to them will be approximate at best. Also, there will always be slight variations as to how a color will be perceived on one computer or another due to ambient light and the computer's video hardware configuration.
About the only practical way to avoid what you're experiencing is to use a color model that's common to every user. Specifically, that means specifying a color either in RGB or Hex (Hexadecimal) values.
Since you deal with commercial printing, you could opt to purchase one of the Pantone color utilities -- but you might not need it. If you have access to Photoshop, you can specify a Pantone color fill then note it's RGB equivalent. There are also several Pantone to RGB conversion charts floating around on the Web. You may consider using a reference chart like this http://www.sandaleo.com/pantone.asp or this http://www.logoorange.com/color/color-codes-chart.php to lookup what you need.