You can use the Nest CLI to update the dependencies:

$ npm install -g @nestjs/cli
$ nest update

You can also $ nest u

As Mick mentioned in his comment, you might have to add --force
argument.

nest update --force

Update – July 7 2022

Since v9.0.0 release, the command update was removed.
To upgrade your dependencies, you can use dedicated tools like ncu, npm update, yarn upgrade-interactive, etc.