Is it possible to power an EF-X500 with a Quantum Turbo or the like?
Probably not with a readily available connector. Third party battery packs such as the Quantum Turbo are usually available with four popular connectors:
- Canon. Also used by a variety of third party flash makers such as Yongnuo, Nissin, etc. for flashes that are Canon TTL compatible.
- Nikon. Also used by a variety of third party flash makers for flashes that are Nikon TTL compatible. Some Nikon compatible third party flashes use the Canon style HV connector, though.
- Sony
- Metz
The connector for the Fuji EF-X500 appears to be a different proprietary connector.
It is also not clear from the information available from Fuji regarding the EF-BP1 and EF-X500 if the Fuji Power Pack EF-BP1 is a true high voltage (HV) power pack or just a higher capacity power pack that feeds 6V to the flash just as the internal batteries do.
HV power packs charge the capacitors in flashes directly, bypassing the capacitor charging circuits between the flash's internal batteries and the capacitor. Battery packs that only increase the total battery capacity do not bypass the charging circuits. If a true HV battery pack is connected to a connector on a flash that is not meant for an HV power source, it could damage the flash's charging circuits that are designed to handle normal voltage from AA batteries.
If the EF-BP1 is the only external power option, is there a TTL flash option for the X-T2 that can be powered with a Quantum Turbo?
Maybe.
The Nissin i60A for Fuji is TTL compatible with the Fuji X-T2 if the camera is running firmware version 1.10. There were some compatibility issues with earlier versions of the X-T2 firmware. The i60A has a power socket that uses the Sony style connector. According to the listing at B&H, it is an HV power socket. If that is, in fact, the case then any Quantum Turbo type of HV power supply with Sony connectors available should work with the i60A.
From the Nissin i60A for Fujifilm Cameras Instruction Manual:

Based in the recycle times listed in the manual, I would verify that the connector on the i60A is a true HV connector before hooking an HV power pack up to it. One would expect recycle times to be in the tenths of seconds with an HV power pack and a flash with the i60A's guide number.