Doom developer id Software is hiring for an unannounced project, which mentions it will be a FPS game much like other games from its library.
gamerant.com
While it's not known if the next game from the developer will be another Quake, recent job postings from Zenimax, id Software's parent company, indicates a new project is at least in the works.
The job posting notes that id Software is searching for an experienced Senior UI Programmer, which is a position that works with the artists and gameplay designers to create player-friendly user interface elements. The position is stated to be for an unannounced video game project, though considering id Software's library, it's very likely to be another FPS game. This is supported by previous job postings from Zenimax that have hinted it is being developed for "long-running iconic action FPS," for example. This can include either another Doom entry, or even a rebooted Quake game.
Other than that, not much is known about this game for the time being. The one thing of note that can be gleamed from this id Software job listing is that it will be aiming for the PC and next-generation consoles. Based on this statement, it's likely that this means the game won't be supported on the past console generation, so it will likely be skipping PS4 and Xbox One.