The answer is no. Googlebot only uses the HTTP 1.1 protocol to crawl websites as per 2018. When you install HTTP/2 on your web server, your web server can still answer HTTP 1.1 request and respond using HTTP 1.1.
Below you can find the HTTP headers response from Googlebot in google search console. The “fetch as google” section of Googlebot helps to know what Googlebot would receive as a response from your web server. As you can see below, for an HTTPS or HTTP request, GoogleBot would receive an HTTP 1.1 response from a web server on which HTTP/2 is installed.
This is confirmed by the logs. In the logs, GoogleBot prompts the web server with a GET request using the HTTP/1.1 protocol. It gets an HTTP/1.1 response as seen above. This means Googlebot doesn’t use HTTP/2. The following screenshot shows a regular visit from Googlebot.
For your information, a HTTP/2 response would look like this:
$ curl --http2 https://tomydna.com -I HTTP/2 200 date: Thu, 30 Aug 2018 20:47:21 GMT server: Apache link: <https://tomydna.com/wp-json/>; rel="https://api.w.org/", <https://wp.me/P9G7UU-d>; rel=shortlink cache-control: max-age=0 expires: Thu, 30 Aug 2018 20:47:21 GMT content-type: text/html; charset=UTF-8