I was planning on doing cable work for my triceps and biceps every second day for the next month.
As a beginner you have a much quicker adaptation to activity then an intermediate or advanced lifter, so to that end this is an appropriate strategy, but with one caveat: time between workouts is not the only factor that affects your recovery. Other factors in recovering include getting 8-10 hours of sleep, eating enough to fuel your increase in activity, eating healthy, staying hydrating, and lifting only what your muscles can support (i.e. don't overload your body). If you do all of these things, then yes, an every other day arm workout would work.
Am I opening myself up to a potential injury by doing this?
Injuries typically stem from one of three issues:
Imbalanced routine: You need to balance out opposing muscle groups that are worked out in order to avoid created muscle imbalances. Since your program balances bicep and tricep work, you aren't at risk here (an at risk example would be someone doing a lot of chest exercises like bench press that doesn't work the upper back to balance it out).
Improper form: Make sure you have proper form in all of your exercises. This is especially prevalent with the bicep curls exercise, where people seemingly always cheat to get extra reps or extra weight that they shouldn't be lifting. Make sure you review proper form for all of your exercises, and use multiple sources for each exercise to make sure that you don't get any bad or incomplete information.
Overtraining: Injuries from overtraining fall into three subcategories: lifting too heavy, getting too little recovery, or exercising too frequently,. The first issue you can avoid by only lifting as much weight as you can do with proper form (and should the form break, put the weight down). The second issue you can avoid by following my instructions in the first paragraph for getting adequate recovery. The final issue is probably what you are most at risk for, and that is something you will just have to listen to your body on. If you find yourself not having enough energy or feeling exhausted, maybe take an extra day off and go to a 3/week routine in the interim.