Quake was the next major game to be released by id Software after the success of Doom. The first release was on June 22, 1996. It is a first person shooter and was clearly influenced by Doom in its style of gameplay, design, and storyline. For example, Quake features a similar arsenal of weapons as well as teleporters and a Satanic theme.
Features original to Quake, such as the console, later appeared in Doom source ports. Once the official Quake source code was released, some Doom source ports incorporated portions of the Quake source. Also, Doom 3's engine is a spiritual successor of the Quake engine (and was used in Quake 4). Another noticeable connection to Doom is that in the later versions the helmet of one of the playable characters is similar to the one on the box art of the first Quake.
Influence of Doom on Quake
The first Quake was built around its engine with no strong story point. The concept for the game itself was divided between roughly three camps: those (notably John Carmack) who wanted a fast-paced futuristic theme, with no supernatural elements for a change; those (notably Adrian Carmack) who wanted a Lovecraftian occult theme; and those (notably John Romero) who wanted a gothic fantasy game centered in melee combat. In the end, Quake went far in none of these directions and instead blended elements of all, resulting in a mix very similar to Doom's own theme: beginning in high-tech maps and moving on to gothic fantasy levels to battle Lovecraftian monsters. The scenario is practically a direct calque of Doom's: once again, the military-industrial complex devised teleportation devices (called "slipgates") which actually pass through a dimension full of hostile monsters who seize this opportunity to invade and must be thwarted, by the sole surviving soldier stationed at the facility. The Quake HUD's layout is likewise closely modeled on the Doom HUD (which itself is not much different from the Wolfenstein 3D HUD).
Many weapons in Quake are similar to Doom's: the shotgun, super shotgun and rocket launcher are directly taken from it.
The Enforcer enemy closely resembles Doomguy, even the death sequence is very similar.
As demonstrated in some Quake alpha screenshots, during the development, some Doom maps were redone for Quake for internal use. 
Influence of Quake on Doom
Quake had a strong impact on the Doom community. The textures from Quake are used in many popular mods (for example, the Torment and Torture series) and Quake-like elements (such as a console or a particle system) have been adopted by many source ports. After the Quake code was open sourced, some ports (most notably Vavoom) borrowed from it.
In Doom 3, one of the PDAs you can get contains a message from the owner's nephew, thanking him for the "Quake 43" game he got for his birthday (An obvious gag to how many Quake games are created compared to Doom games)
In the 2016 reboot of Doom a weapon called the "Lightning Gun" that 'disperses a constant stream of electricity to the target" appears, an obvious parallel conjunction to the Thunderbolt from the original Quake.