Nest runs its API over HTTPS, which uses port 443. Even if they allowed port 80 you should not use that; you'd be transmitting your credentials or API token unencrypted, which is dangerous.
If your code is running on an ESP8266 be sure to use an SSL library. It's not enough to just speak the HTTP protocol to port 443.
You didn't share any code so I can't advise further, but consider using the BearSSL::WiFiClientSecure class for your connection.
There's a good example of this at
https://github.com/esp8266/Arduino/blob/master/libraries/ESP8266HTTPClient/examples/BasicHttpsClient/BasicHttpsClient.ino
You'll need to know the SSL fingerprint for the Nest server. You can find this by running
openssl s_client -connect api.home.nest.com:443 < /dev/null 2>/dev/null | openssl x509 -fingerprint -noout | cut -d'=' -f2
in a terminal window on a Mac or Linux computer. The fingerprint allows your client to confirm that the server is who it claims to be.
As I write this, the fingerprint is currently:
DE:AA:EB:EE:C0:4B:14:97:27:C8:29:46:5C:05:44:2C:26:DE:55:6B
Be aware that this can change with time (or even depending on the specific server which handles your request), so you may need to change it. You can find the line that calls client->setFingerprint() in the example at the link above; that's where you'd use server fingerprint.