Short answer. No. The Labor Code says no notice to the employer is required at all if the contingency is known to the employer or his agents or representatives. Telling a supervisor puts the knowledge where the law wants it, and the article does not prescribe a form for the notice.
What the law says
No notice to the employer shall be required if the contingency is known to the employer or his agents or representatives.
Labor Code, Article 206 — Notice Of Contingency. Read the full provision →
Knowledge substitutes for notice
The purpose of notice is to let the employer learn of the contingency in time to record and investigate it. Where he already knows, the purpose is spent, and the article says so directly by dispensing with notice altogether. It extends that to knowledge held by his agents or representatives, which is why a supervisor matters — a person exercising authority on the employer's behalf is not a bystander, and what reaches him reaches the employer for this purpose.
The article prescribes no particular form
Read the notice requirement itself and you will see what it does not say. It requires that notice be given to the employer by the employee, by his dependents, or by anybody on his behalf, within five days from the occurrence. There is no requirement of a prescribed form, a written report, or a company template. Company practice may ask for one, and it is sensible to comply, but a house rule about paperwork is not the statutory condition your claim has to satisfy.
Anyone may give the notice for you
The breadth of or anybody on his behalf is deliberate, and it exists because an injured worker is often in no state to make reports. A spouse who phoned the office, a co-worker who brought you to hospital and told the shift head, or a dependent who informed HR all satisfy the article. So if you were unconscious, hospitalised, or simply too badly hurt to attend to formalities, the notice question is answered by what anyone did on your behalf, not only by what you did yourself.
Turn the conversation into a record now
Knowledge is only useful if it can be proved later, and a verbal report leaves nothing behind. Send a short message or email to the supervisor confirming what you told him and when — even weeks afterwards, a message referring back to the earlier conversation is far better than nothing. Note who else was present, keep your medical records and their dates, and ask whether the incident was entered in the company logbook, since that entry is the employer's own record of the event.