Not only would I expect the retention times to change, but also the peak areas too. Thus, the fingerprint will be different. However, the detector response will change for each peak too, so the %mass would not be significantly different.
"Does changing column in GC-MS effect the method for analysis?"; Most certainly! The column is part of the method. Change the column and you have changed the method so it is no longer select for the sample. The two GC columns are different. Columns are selected for very specific reasons.
As you are brand new to chromatography, please take the time to learn the basics of gas chromatography (read up on before using) and get some local help. In doing so, you will begin to understand what a "chromatography method" is and how the parts that comprise 'a GC method' are responsible for and determine the outcome (the results). *Learning about new analytical techniques will greatly help you with other projects too.
Dont forget that GC column bleed could affect the background and noisy relationship in peaks, thus you should consider that the replacing column affect the sensivity even in the same brand column and reference.