I figured it out. The Google Analytics API uses different filter formatting for GET and POST methods. In my example, I am using the GET method with the Analytics::performQuery(). I found the documentation for the proper filtering here. After playing around with the code, my new filters look like this:
$period = Period::days(7);
$metrics_total = 'ga:pageviews';
$metrics_unique = 'ga:uniquePageviews';
$dimensions_results = [
'dimensions' => 'ga:pagePath',
'filters' => 'ga:pagePath==/results',
];
$dimensions_home = [
'dimensions' => 'ga:pagePath',
'filters' => 'ga:pagePath==/',
];
$total_results = Analytics::performQuery($period, $metrics_total, $dimensions_results);
$unique_results = Analytics::performQuery($period, $metrics_unique, $dimensions_results);
$total_home = Analytics::performQuery($period, $metrics_total, $dimensions_home);
$unique_home = Analytics::performQuery($period, $metrics_unique, $dimensions_home);
where the 'filters' element of the $dimensions array indicates what is being filtered, the operator, and the condition. In my case, the filter is if a page path equals a certain string. Spatie/laravel-analytics takes care of the URL encoding required, so I can type it in plain text like above. The result returns a collection instance containing the Google Analytic data requested.