To use your own search box in Google’s custom search, you must create a custom search engine. Go to "Appearance" and select the "Layout" tab". Choose the "results only" option and then click "save and get code" You will be taken to a page where the code is that you should paste into the font of the page where you want the results to appear on your site. There is a "Specify Search Results Details" button. By clicking this button, there will be a field with the parameter to pass the query to the query, which by default is "q". This parameter must be equal to the value of the "name" attribute in the text field of your search box. The "action" attribute of the form in your search box must be equal to the address of the page where you want the search to appear, that is, where you placed the previously mentioned code.
<form method="post" action="pagina_de_resultados.html">
<input type="text" name="q" value="Pesquisar...">
<input type="submit" value="Pesquisar">
</form>
You have to be more specific, if possible post some prints, the code and describe what you expect to do, the result to be obtained, and without worrying about how you will get to it, because in this we can help you.
– Clayderson Ferreira
Is the site completely HTML? I believe that to create a search system, you will need to improve the backend of your site, that is, use PHP/ASP/ASP.Net...
– Chris Born