The quick answer is "it depends."
If your computer speakers already feature a sub (I assume this is your situation) and you're looking to simply replace it using the existing amp, I see the following potential problems:
1) Impedence mismatch: your sub is probably 4 ohms (or maybe even 2 ohms) per coil. Your computer speakers' sub amp might expect a much higher impedence (My Definitive Technologies home theater speakers use a pair of 30 ohm subs wired in series for a total of 60 ohms!). Trying to attach a your Alpine sub to such an amp will damage the amp.
2) Will the existing amp deliver enough power?
3) Frequency response. The Alpine sub specs I've seen were rated for a pretty wide frequency range (upwards of 1000Hz) but I don't know how well they really perform above, say, 150Hz. You might wind up with a speaker setup that offers good bass, good treble (from your existing satellites) but not much mid-bass. Dunno. Depends on where the computer speakers crossover is set (if there is one).
4) Shielding. Probably safe to say your Alpine sub doesn't have a bucking magnet so you'll probably need to keep it a fair distance from your machine and monitor. Not that this is a problem, just something to keep in mind
If you have a spare amp and crossover laying around, you could just split the line-level audio signal coming out of the sound card and send it to your alpine's amp. 'Course if you had that laying around, you'd have probably tried it.