I have recently signed up for HubSpot and have connected my G Suite account to it. However, when I click on a contact, and go to the Emails tab, I do not see any emails sent to received from that contact. I did not send emails through HubSpot, sent them through G Suite in Chrome.
To be able to log emails in HubSpot from your connected email account, you need to install the HubSpot Sales extension/add-in. As you're using Gmail and Chrome, you'll need to install the HubSpot Sales Chrome Extension. The steps can be found here.
In the below article, you'll have a good starting point on how the extension is working: