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For some time now I have had the desire to carry out this action. I am looking for a way to integrate the forum menu, as shown in the image, into my WordPress page. Although I have already linked all the necessary elements, I am still looking for a way to accomplish this. I would greatly appreciate any help, as I have been researching on the internet with no success so far.

Sorry i do not have understand the question.
Do you mean the template integration?
https://www.axew3.com/w3/2020/01/phpbb- ... iframe-v5/

I mean to make only the forum menu or the forum header appear as shown in the image.

Well, thinking to how to do this with easy, i could think to the shortcode template integration, that could be quite easily instructed to display only the top of the forum when required for the shortcode instance used: do you mean all the forum top header as it is on your screenshot, or just the links?
The first via template iframe shortcode is quite easy (i say via shortcode so to add it everywhere into template page)
https://www.axew3.com/w3/2022/02/the-aw ... shortcode/

File
wp-content/plugins/wp-w3all-phpbb-integration/views/wp_w3all_phpbb_iframe_short.php
you can see that changing this line: into this (giving css height and max-height, into the inline style attribute: 180px height into the example) So it can be easily added as shortcode param option into the default Awesome Template integration by shortcode (will be added on next coming 2.8.3) then used to display the iframe instance with the needed height.
The result will be this:

I would like to make a suggestion. I find this system amazing because I personally find it very useful, but I have faced a problem: I'm not sure if I have correctly integrated everything as described here: https://www.axew3.com/w3/2020/01/phpbb- ... iframe-v5/

I think it would be very helpful if you could create a detailed video tutorial showing step by step how to carry out this procedure. Both myself and other users could benefit from such a guide, as many of us might have difficulty understanding the process correctly in writing.

I hope you can consider this suggestion so that more people can take advantage of your tool, which is really helpful.

I will do an install today asap, explaining detailed short step by step, in a better way and very simple!
You may don't believe that I was thinking the same last night, while coding a new feature: that updates phpbb last topics on widgets, or notifications, without to have to reload WP. The code will be released as ready (it is fully working on my tests)

While the step by step, fast, iframe template integration guide is coming today. Ok!