As you want to get the details of dates and from where you can get Merchant Navy entrance exam forms so here is the information of the same for you:

Its forms are available perhaps in the month of December

You can get the details of update information about it by following regularly the employment news paper.

Here for your reference I am giving you the list of some of the best Merchant Navy training colleges in India:

Mercantile Marine Academy Foundation
Agnel Maritime Academy
C.V. Raman College of Engineering
Coimbatore Marine College
Marine Training Academy
Maritime Academy of India
Tolani Maritime Institute
International Maritime Institute Ltd
Indian Maritime University (Kandla Campus)
Indian Maritime University (National Maritime Academy)
Oceanic Maritime Academy
Seascan Marine Services Pvt Ltd
Amer Maritime Training Academy
Marine Education Charitable Trust
Kolkata Port Trust
Yak Educational Trust
OERC Academy
MASSA Maritime Academy
International Maritime Training Centre
Perunthalaivar Kamarajar Institute of Maritime Science & Engineering
Kunjali Marakkar School of Marine Engineering of Cochin University of Science and Technology.
Sea Explorer's Institute Educational Trust
Centre for Maritime Education and Training
International Marine Academy
Fees - Centr for Maritime Education & Training
Fees Balaji Seamen Training Institute
Fees - HIMT Post sea courses
Mangalore Marine College
Univan Maritime Training Academy
Seafarers Marine Institute
Institute of Marine Education & Research Pvt Ltd
United Marine Academy
Subbalakshmi Lakshmipathy College of Science
Maritime Training Institute
Ramana Academy of Maritime Studies
Anglo Eastern Maritime Training Centre
SCMS Maritime Training Institute
Pentagon Maritime Training & Research Institute
Cochin Shipyard Limited
Bonzer Academy of Maritime Studies
Haldia Institute of Maritime Studies & Research
S.B.Vignesh Marine Training Centre
National Inland Navigation Institute
Indian Maritime University (TS Chanakya)
Institute of Marine Engineers (India)
College of Maritime Studies & Research
Naval Maritime Academy West
MASSA Maritime Academy
AMET University
Noorul Islam University
BP Marine Academy
Mangalore Marine College (MMC) [Mumbai Campus]
MMTI's Education & Research Trust
Sensea Maritime Academy
School of Maritime Studies (Vels University)
Chennai School of Ship Management
ABS Industrial Verification (India) Pvt. Ltd.
RL Institute of Nautical Science
TS Rahaman
Institute of Maritime Studies
Indian Maritime University, Kolkata Campus [Indian Institute of Port Management]
Indian Maritime University (LBSCollege of AMS)
Institute of Marine Engineers (India)
Gurship Education Trust Maritime Training Institute
Indian Maritime College
Indian Maritime University (National Maritime Academy)
Mumbai Maritime Training Institute
Setrac College of Offshore Training
Centre for Maritime Education & Training
GKM Institute of Marine Sciences & Technology
Sagar Gyan Academy
Applied Research International
Marine Medical Clinic
Marine Training Academy
Indian Maritime University (MERI)
U. V. Patel College of Engineering
CMC Maritime Academy
Indian Register of Shipping
GKM College of Engineering & Technology
Anglo Eastern Maritime Academy
Indus Seafarers Training Academy
Samundra Institute of Maritime Studies
St. Xavier's Technical Institute
Garden Reach Ship Builders & Engineers Ltd
Hindustan Institute Of Maritime Training Post Sea Training Centre
Suraksha Marine
Fosma Maritime Institute and Research Organization
Arya Marine Academy
Natcom Education & Research Foundation
Aquatech Institute of Maritime Studies
Yak Management & Marine Education Centre
HIMT College Hindustan Institute Of Maritime Training Pre-Sea Training Centre
Indian Maritime University (MERI)
SNS Maritime Training Institute
Southern Academy of Maritime Studies
Anglo Eastern Maritime Training Centre
Cosmopolitan Technology of Maritime
Fees - HIMT Pre sea courses
Institute of Petroleum Safety, Health & Environment Management
Euro Tech Maritime Academy
Mazagon Dock Limited.
Chidambaram Institute of Maritime Technology
School of Seamanship and Nautical Technology
Mohammed Sathak Engineering College
Orissa Maritime Academy
Univan Maritime Training Academy
J. Sons Merchant Navy Institute
Maritime Foundation
Sriram Institute of Marine Studies
Francon's Marine Academy
International Maritime Academy
Fleet Management Training Institute
Mumbai Maritime Training Institute
Fees International Maritime Academy
Oceans XV Educational Trust
Seacom Marine College
Balaji Seamen Training Institute
SeaSkills Maritime Academy
Don Bosco Normar Maritime Academy
School of Maritime Studies (Vels University)
Park Maritime Academy
NUSI Maritime Academy
School of Synergic Studies