Someone referenced my work in a journal paper using my first name only, do I contact them or the journal it was published in? Someone referenced my work in a journal paper but they did not cite it the right way.

They only included my first name, my supervisor's first name, the name of the paper and the date.  They did not include our last names or the journal that the paper was published in.
Do I contact the authors or the journal to correct this? Or can it not be fixed? 

Currently, the citation is not included in my Google Scholar page because they didn't write my last name nor my supervisor's last name.
I would recommend approaching this problem through the following layers of triage:


First, is it a serious journal, or a predatory trash publication?  If it's not a serious journal, just ignore it and move on. It won't really count as a citation anyway, and there's basically nothing you can do to get a meaningful response from the people running it.
Second, if it's a real journal, is it in the final form or a "just accepted" pre-print?  Some journals give early access to "just accepted" manuscripts before any copyediting occurs, typically with a big warning label saying "Not yet final, things may change."  If this is the case, then you may be able to count on the production staff catching the problem and getting it fixed before the final form; if you don't want to count on the error being caught, it is reasonable to (politely) contact the authors.
If neither of these is true, and the paper is in its final form in a real journal you can write to the editorial staff to request a correction.