Documents to be collected before I quit a job

I work for a company in Japan. I have resigned now with 1 month notice period. What are documents need to be collected before I leave. I am going to join another company in Japan.

Ask your new company what they need.
You will have your answer than

Documents that are related to your taxes, health insurance, pension that will need to be transferred to your new company.
Also get hold of your Blue Pension Book if you didn't got it from your company (your 1st job?)
As other said, double check with your new company HR on what is needed.

You may or may not get certificate of employment, which details your work history in the company.

Also be careful if you have gap in between last day of old work and start of new work.
The national health insurance and pension may ask you to pay premium for coverage during those gap. You 'should' register yourself at city hall for coverage during those gap time.

Good read on Leaving for another job or between jobs?
https://origami-book.com/column/course-en/10658

Don't forget to inform the immigration about the changes in your employment status within 14 days.
You can do it online
https://www.ens-immi.moj.go.jp/NA01/NAA01SAction.do;jsessionid=005FFDA...

or by postal
http://www.immi-moj.go.jp/english/tetuduki/kanri/shyorui/19.html
.

I am last working day in current company is 15th Feb. And first working day in new company is 5th March. Still, do I need to inform the immigration office about my status?

Yes, you always need to inform Immigration of your change in employment. Within 14 days of quitting and 14 days of starting a new one.

Also try to get your 退職証明書 too if you can. I needed it to renew my visa.