Well, after a lot of research, I found the answer. Apparently, Facebook is no longer accepting GET calls without using an access token. I ended up using the same URL of the above question but adding some parameters. So, it was like this:
https://graph.facebook.com/v3.2/?id=URL_DA_PAGINA&fields=engagement&access_token=TOKEN_DE_ACESSO_DO_APLICATIVO
I used the access token of an application I had already created on Facebook Developers and I redid the total URL. JSON generated by Facebook is now something like this below:
{
"engagement": {
"reaction_count": 70,
"comment_count": 52,
"share_count": 61,
"comment_plugin_count": 0
},
"id": "http://www.endereco-da-pagina-pesquisada"
}
To capture the data, then, you need a call like this within Javascript:
var MeusAcessos = parseInt(data.engagement.share_count);
This solved my problem. In case someone reads this answer and needs more information, post your question.