Any version of DisplayPort will support 1920x1080 @ 144 Hz. You will need at least DP 1.2 for 2560x1440 @ 144 Hz.
If your laptop does not have a DisplayPort output, it likely does not have a GPU capable of rendering over 60 FPS anyway.
You can get external graphics adapters for laptops. You would have to look at the USB or Thunderbolt connections available on your laptop for connection.
https://www.gigabyte.com/Graphics-Card/GV-N1070IXEB-8GD#kf
Edit:
You need a thunderbolt 3 connector for these external graphics adapters, which many laptops do not have.
If the GPU in your laptop is suitable for games over 60 FPS then the laptop should already have a DisplayPort connection.
Outside of games, you will get no benefit from running a screen at over 60 Hz.
HDMI 1.3 and higher supports 1920x1080 @ 144 Hz and HDMI 2.0 and higher supports 2560x1440 @ 144 Hz. These are an alternative to DisplayPort, but you need to ensure support by both laptop and monitor and get a compliant cable. Simply supporting HDMI isn't enough.