Alright so it's really about mixing and matching and finding what looks and sounds good to you. I will provide translations for both words to choose from, and then show you how to combine them.
Conquer
- vincere (this reflects Vincent nicely (as it is the origin of said name), so the next option is just for kicks)
- domare (as in dominate)
Defend
Unfortunately, Alexander is Greek in origin, so there is no Latin verb from which it derives, so these are all just translations of "to protect."
- servare
- tegere
- tutare
- munire
- arcere
Some options have been omitted due to length.
Putting it Together
Now, as in English, one would normally join two clauses like this with a conjunction such as "and." Latin can do the same thing, and provides two options that are pretty common (even in English).
- et as in VINCERE ET ARCERE (This one is most familiar to non-Latin English speakers, as it appears in common phrases such as "et tu, Brute")
- -que as in VINCERE ARCEREQVE (The -que is a suffix of the second word)
You could omit a conjunction as well, to reflect the English example you gave ("CONQUER DEFEND" -> VINCERE ARCERE).
You will also notice that I capitalized all of the letters in these examples and replaced any u's with V's. This is how the Romans would write inscriptions on monuments and such (see here), so it makes sense to me to do it like that on a tattoo, especially if you use a similar font.
You may have noticed that I used arcere in all of my examples. I like this one the best especially when paired with the -que conjunction because of the consonance with the c's and q (even if it becomes somewhat of a tongue twister!). However, it is up to you to choose the words that you prefer best, as well as the conjunction. I hope this assists you, and good luck!
If anyone knows of any verbs that derive from the Greek words that were the original source of Alexander, I'd be very interested to hear about them.
