This is definitely doable with handlebars, but a few things to consider:
page.url
is an absolute URL – so this approach could work, if this is for a Ghost site that you know the static URL for (so not for a theme you’d use on multiple different sites, for example).
Instead of the #if
helper, we need to use the #match
helper. #if
tests very simple conditions, whereas #match
can do proper comparisons. Here’s more on the helper.
So, combining these two bits, we’ll end up with this code (replace http://localhost:2368
with your domain):
{{#match page.url "=" "http://localhost:2368/about/"}}
html code
{{/match}}
An alternative, and in my opinion more flexible approach, would be the following:
{{#post}}
{{#has slug="about"}}
html code
{{/has}}
{{/post}}
We need the {{#post}}...{{/post}}
snippet to check whether we are in a post context. A post context also stretches to pages, so all we’d need to make sure is that the URL we want to check is either a post or a page (if you expect it to be a tag or author page, you’d need to change the context accordingly).
Then, once we’re sure that we’re in the right context, we use the #has
helper to check if the slug of the page is “about”. Yeah, yet another helper – but pretty useful in this case. Here’s more on it.
Personally, I’d prefer the second approach to solve this, as it is versatile and doesn’t rely on the absolute URL 
Hope this helps!