Nice guess :) As noted in the comments, はじめよっか is basically a shortened version of はじめようか.
Also, more than generalizing about that type of phrasing... I'd almost want to say it's more of just a way to make a phrase sound more "clipped" (e.g. something of a glottal stop, or possibly, a contraction...) or maybe even just an alternative way to put an accent or stress on a part of a phrase... as opposed to something grammatically formal or defined.
For example, the phrase "getting good" in English is sometimes phrased as "gettin' good" (in this case, read as "get-tin' good".) The syllable count is the same for each phrase... but the latter phrase can potentially sound more clipped (see glottal stops) though this example is not necessarily an exact match to the ようか / よっか change.
Usage note: you may want to be careful with using that style of speech unless you are older than the person with whom you are speaking. (Or if you are somehow related... or are really familiar with the person.)