There is no such thing as a standard FileMaker Pro Note field. There is such a thing as a text field. It is possible to set up a script trigger that will produce the same result as a "clickable link", with the right scripting, text of a specific style in the text can be used as input for a script--resulting in navigation to another part of the database, opening a URL in the default web browser or displaying that URL's page in a web viewer (and many other options are possible).
I'm sorry, in the Contacts.fmp12 template there is a field Notes that does not allow a clickable link. I would assume that text container would present a clickable link when a correct link string is pasted into it. Without the need for other handling. I see if I highlight the link text and right click I can Open the link now in a browser. But that is too many steps.
The Field Notes is a text field. If you added some scripting and script triggers, a portion text styled (say blue and underlined like a typical hyperlink) could function as a hyperlink. This works in FileMaker Pro but not in FileMaker Go on an iOS device.
Clicking the field can trip the OnObjectEnter trigger. The script can check the text style of the character at the location where the click put the cursor and can use a pair of loops to scan left and right of the point clicked to find and extract the the entire sub string with this style. It can then use Open URL to open a web site.
But it's simpler to set up a text field only for entering a URL, then clicking that field or a button associated with it can use Open URL to open the web site without needing to parse the text from a field containing other text.
A URL entered into a text field can also be the text used to display a web site in a Web Viewer.
For my uses, I'm unlikely to set specific URL fields for some of this adhoc contact data that may contain URLs and be brought into the Notes Field on import.
Not everyone in my contact list would share some of these specific URL needs, and they vary, i.e. book link, movie link, story link, flikr page etc..., so I see no reason to make specific fields to house that data now. And now that you mention and I see that FM Go doesn't support a link in a Note field I'm still not impressed.
The App Evernote for IOS takes any data in one of its Notes and renders URLs in its own browser, PDFs in its own viewer etc.. No need to open a 3rd party app or cut and paste. Must I build a Contact DB in Evernote and skip FM?
In FM GO I'd love to be able to use search to pull out attributes from picklist values in multiple fields. i.e. show me all people who play golf and who are in xyz area codes (415, 650, 408) in northern ca, or better home cities but more of them. If only Siri could do that!
Yet a portal of such URL fields can do exactly what you requested. And with a portal to a related table a given contact can have o links to many links.