I have created a free website at wordpress.com and I want to get it indexed in Google's search engine. I came across a video saying, you need to verify your website in "Google search console" in order to index it in Google engine.
The problem is that the free version of wordpress.com does not allow me to use the "Header and footer plugin" in order to put HTML tag in it and verify my website. Are there any alternative methods?
First, I'd like to caution you against using wordpress.com's free plan. With their free plan, you can't use your own domain name. When you are ready to move off their free hosting and want to redirect your "free" site, your only options would be to pay $13 per year to enable the redirect or use a poor man's redirect that doesn't preserve your SEO and makes users click.
People often hear the advice that they should get a "WordPress website". What that really means is a website with the WordPress content managament system (CMS) available from wordpress.org installed on it. It doesn't mean a site hosted with the (relatively expensive) hosting packages at wordpress.com.
I would suggest that you buy a domain name and search for cheap shared hosting. If I Google "cheap wordpress hosting" it looks like you can get plans for $10 a year. With domain name registration and hosting you can certainly get started for as little as $25 to $30 per year.
Building a site on somebody else's subdomain is only going to lead to tears.
Second, Google Search console doesn't actually help your site get indexed. Googlebot will be able to crawl and index any site created with WordPress.com. Creating a Google Search Console property won't help with that process. Google Search Console gives you insight into how much of your site is getting indexed and which keywords people are searching for to get to your site.
Wordpress.com has the instructions here. The basics are:
You don't need a special header and footer plugin to use the HTML meta tag.