You can replace the speakers to run off the factory Bose Head Unit, however you may experience lower volume due to the factory speaker impedance mismatch. Nearly all factory speakers have an odd ohm rating, often times 3 ohm speakers. The Alpine speakers, like 95% of speakers out there, are 4 ohm speakers. In order to get the best sound, its best to drive them with an external amplifier. In this case, you would need to run new wires from your amp to your speakers. Doing this will improve your sound GREATLY, sound quality wise and volume. You can try it out and see if it's enough volume for you.
My Armada came with MA Audio speakers in the doors and the speakers in the dash are the factory Bose speakers. My door speakers are driven off of an old school Planet Audio amplifier. I will be soon replacing the MA Audio speakers and the dash tweeters with my Focal Utopia 6.5" component speakers that I have had for the past oh, 6 or so years. The rear will be replaced with either my Focal K2P with Digital Design T1 silk dome tweeters or my Focal Performance PS165 6.5" components. Both will be bridged off of a 4 channel amp each as the Focal speakers can take some serious wattage.
I GOT MORE SPEAKERS THAN YOU. 2004 Armada SE 4x4 Fully Loaded, Bilstein 5100's, 2" Rear Lift, Towing Package, Leather, Power Seats, Auto-Air, Rear Parking Sensors. Rockford 3Sixty.1 DSP, JVC KW-NT800HDT, FRONT: Focal Utopia Comps powered by Audiopipe AQX-360.4 Bridged @ 320wrms x 2, 2nd Row: Focal K2p Comps powered by Bangs & Olfsen ZED Audio 4 Channel Amp Bridged @ 275wrms x 2, Center Channel powered by Phoenix Gold Tantrum 500.2 @ 175wrms x 2, FOUR RARE Sundown Audio SA-8 V.2 w/ SSA internals, 3cuft ported box @ 31Hz, 2 Audiopipe AP18001D's 1800wrms x 2 (3600wrms) XS D3400 Battery. 4800wrms