Yes, you can certainly buy just one share of Google (or any other stock, for that matter).
While the price of a single share may seem high at first glance - currently around $1,200 per share - keep in mind that you can buy fractional shares of most stocks. This means that you can buy, say, 0.25 shares of Google for $300.
If you're interested in buying Google stock, you can do so through a broker like Vanguard, Fidelity, or Charles Schwab.