No one is perfect, but IMO when it comes to security, Google goes to great lengths to make stuff in the web secure, probably the most from all the bigcorps out there.
They are very aggressive about fixing TLS/SSL-related issues (finding and fixing vulnerabilities, deprecating old ciphers, promoting new stuff like cert pinning and HSTS, lowering SERPs in Google for non-https websites; they've also put in place means to detect and report fraudulent HTTPS certs, and many more).
Many of the top security researchers work for Google, and many of them even not on things directly related for Google's daily businesses.
If I'd have to say about some bigcorp that they don't care about security, Google would be the last on this list.
Google customer support is totally another pair of shoes though.