I have just tried to reproduce the issue on a local test server and I was able to create a set of tabs with a form in each tab with no issues.
Check that you have entered the shortcodes correctly.
The shortcodes I used were:
[tabby title="Tab One"]
[contact-form-7 id="178" title="Contact form 1"]
[tabby title="Tab Two"]
[contact-form-7 id="179" title="form 2"]
[tabby title="Tab Three"]
[contact-form-7 id="180" title="form 3"]
[tabby title="Tab Four"]
[contact-form-7 id="181" title="form 4"]
[tabbyending]
If you copy these into your text editor, don’t forget to change the CF7 parameters to match the id of each of your forms.