Apparently, no
The Star Trek: The Next Generation Technical Manual (considered, if not entirely canon, at least a quasi-official authoritative source), gives the following description of the Captain's Yacht:
The yacht is capable of sustained sublight velocities approaching 0.65c. The impulse propulsion system (IPS) consists of six sequential beam-fusion reaction chambers feeding a central toroidal driver coil. Each chamber measures 1.3 m in diameter and is similar to its larger IPS cousins on the Enterprise. The reaction exhaust is vented through the driver coil and magnetohydrodynamic (MHD) accelerator for impulse travel. The MHD tap provides power to the navigational deflector grid for removal of interstellar dust and gas from the vehicle flight path. 126
For atmosphere flight the exhaust is redirected after it exits the MHD tunnel and sent through the aerodyne engines around the ellipse equator. The normal atmosphere cruising velocity is Mach 6; maximum safe waverider velocity is Mach 20. Magnetic turbulence contour equalizers, variants of the navigational deflector, provide momentum conditioning at Mach transitions.
The Enterprise is designed to operate safely without the yacht in place, since its structural integrity field and inertial damping fields produce slightly modified fields in those areas to compensate for the concavity of the yacht docking structure. Yacht operational rules in the vicinity of the starship are generally the same as those for other auxiliary spacecraft, with the difference being that the yacht, during emergency undockings, may be safely deployed at velocities as high as Warp Factor 7. The yacht's systems are designed to afford the craft a smooth falloff of warp field, though the decaying field energy cannot be sustained for any appreciable time.
It explicitly states that the Yacht is equipped with an Impulse Engine, but doesn't provide any information about a Warp Propulsion System, seemingly implying that it has none; also, the statement that it can be detached from the starship traveling at Warp speed and perform a "smooth falloff of warp field" seem to further imply that it can't sustain warp speeds on its own.
On the schematic that is present on the Manual, some systems are detailed (including the Impulse engine), but also here there is no mention of a Warp engine: 35