Troubleshooting HeidiSQL with initial set up (two different Postgres servers in two different networks)

With two different Linux servers I have installed Postgres on them using no standard configurations. They are in different networks off-site.

The Postgres service is running (as I checked by running ``` ps -ef | grep postgres)

These commands show port 5432 is not blocked:

nmap -p 5432 x.x.x.x

I can SSH to either Linux server from my Windows machine.  I have Heidi SQL on my Windows desktop.  When I try to connect to either of these servers with HeidiSQL (using port 5432), I get "could not connect to server: Connection refused...is the server running on host "x.x.x.x" and accepting TCP/IP connections on port 5432."  This has never worked before.

What should I do to get HeidiSQL to connect to a Postgres database on a Linux server?