How to add a hyperlink in a Google Docs using a Google Script

To add a hyperlink in a document use Body.appendParagraph with setLinkUrl, then merge.

let doc = DocumentApp.create("My Document");
let body = doc.getBody();
body.appendParagraph("Please click ");
let link = body.appendParagraph("here").setLinkUrl("http://www.google.com");
link.merge();
let closing = body.appendParagraph(".");
closing.merge();

The code above will create a document with text that looks like:

Please click here.

You should be able to use setFormula and the Hyperlink formula like so:

var value = '=HYPERLINK("www.google.com", "Google")';

SpreadsheetApp.getActiveSpreadsheet()
   .getSheetByName("Sheet1")
   .getRange("A1")
   .setFormula(value);

Edit: Looks like I misread the question. Try this instead:

DocumentApp.getActiveDocument()
  .getBody()
  .editAsText()
  .insertText(0, "link text")
  .setLinkUrl("www.google.com");

Edit 2: Looks like .setLinkUrl() is effecting the whole body, not the text inserted. If you put the link text into a variable and use the length of the variable to mark the link area, it should work. Try this instead:

function insertLink() {
  var text = "link text\n";
  var url = "www.google.com";
  DocumentApp.getActiveDocument()
    .getBody()
    .editAsText()
    .insertText(0, text)
    .setLinkUrl(0, text.length, url);
}