Can we connect 2 cat6 cables as a input to a cat 6 jack

So at the place of wireless router I have socket with only one cat 6 jack. So I will be connecting the cable from the internet modem to this jack as input. And then I need to run the cable to the other room as well. So I was wondering whether I can join these 2 wires by connecting both as the input for the jack.

No, Gigabit Ethernet does not allow 3-way connections. It needs direct point-to-point connections, i.e. one jack—one cable. If you need to connect multiple devices to a single Ethernet port, use an "Ethernet switch" which will give you more ports.

(Especially if you spent extra for Cat6 cable... The sometimes suggested passive hub trick would limit you to 100 Mbps at most, probably even less than that in reality. Don't create potential future problems.)