How to find out if my Apple Watch series 4 can be connected to my phone carrier

See here for supported carriers: Apple Watch - Carriers - Apple

If your carrier is listed there, then it can support your watch.

Note that the watch must be the GPS+Cellular models. GPS only models cannot be connected to a cellular carrier.

Additional to the comments from Phil0124, Apple Watch LTE versions have specific hardware models for different cellular regions. If you purchased in the country you wish to use the watch that should not be a problem, but if you purchased in a different county it may not be compatible with your home country networks. In particular we see many problems reported by users who purchased while visiting USA (watch works in USA, Canada, Mexico) then getting failures when they get home to other countries. This is a hardware limitation and there is no way to work around or upgrade the watch for compatibility.

Call your phone carrier they will know.

And you can easily tell if you have a cellular model, because it will have a "Red" dot or ring on the Digital Crown. You can also look in Settings - General -> About