While I have not seen the paper referenced in Garry Walker's answer, most descriptions I have come across suggest the answer is "no".
The ship inside the bubble of an activated Alcubierre Drive is in free fall, and isolated from any external influences outside of the bubble. The crew of the ship might have artificial gravity by rotating the ship or having a rotating hab as part of the ship, but there is no gravity operating inside the bubble, and external gravitational, radiation or even photons of light cannot penetrate the bubble, nor anything inside escape to the outside universe.
This isn't "anti gravity" in the sense that we think about it (and the use of "negative matter" is similar to the idea of using struts or bracing to hold the bubble in place, except that normal struts or bracing made of ordinary matter would not have a way to interact with the warped space of the Alcubierre Drive). In fact, to use SFNal analogies, this is more like a "cloaking device" (the ship is invisible to any form of radiation, and is not radiating into the outside universe, so any heat or energy signature is hidden). The other thing the Alcubierre Drive can be compared to is a "Sheild", since there is no way for energy to pass into the bubble and the ship inside, so even gigawatt laser batteries would make no impression on a Alcubierre Drive field.
In "Star Trek" terms, the Alcubierre Drive is actually the "warp drive", shields and cloaking device all in one step, once the device is activated. Sadly it is not an anti gravity device.
Negative matter may (according to some theories) seem to act as an anti gravity device, but that is because things are "reversed" (i.e. positive matter attracts but negative matter repels, if you push on a ball of negative matter it accelerates towards you and other bizarre effects). Some clever engineering of negative matter could simulate "anti gravity" (for example a floating platform), but making it useful might be a challenge (you place a mass on the platform but the platform begins to rise skywards in proportion to how much mass there is and how fast you dropped it on the platform...)