Hi, Tony,
I have contacted Tony's mother [Tony's mother is a proper noun so you can't say the Tony's mother], I have contacted the Nurse's mother [The Nurse's mother is a common noun, so you can use "the" before the noun].
"Tony" is a proper name, and "nurse" (no capitals) is a common noun. When used in the genitive case, you should in fact say "Tony's mother" and "the nurse's mother."
I went to the Fiji Mountain in Japan before Covid hit us. I believe the Fiji mountain is a proper noun, comparable to Tony's mother but why it sounds okay to use the article "the" in the case of a Fiji Mountain?
This is not an example of the possessive case. The rules that determine the use or omission of "the" are those that apply to geographical names.
On page 142 of Swan's Practical English Usage (Fully Revised International Edition, 2016) we can read:
18 place names
We use the with these kinds of place names:
• seas (the Atlantic)
• mountain groups (the Himalayas)
• island groups (the West Indies)
• rivers (the Rhine)
• deserts (the Sahara)
• most hotels (the Grand Hotel)
• most cinemas and theatres (the Odeon; the Playhouse)
• most museums and art galleries (the British Museum-, the Frick)
We usually use no article with:
• continents, countries, states, counties, departments, etc (Africa, Brazil, Texas, Berkshire, Westphalia)
• towns (Oxford)
• streets (New Street, Willow Road)
• lakes (Lake Michigan)
Exceptions: places whose name is (or contains) a common noun like republic, state, union (e.g. the People’s Republic of China, the United Kingdom, the United States). Note also the Netherlands, and its seat of government The Hague. There are a few countries whose names used to have the, but are now normally used with no article: (The) Ukraine, (The) Lebanon, (The) Gambia, (The) Sudan.
The is unusual in the titles of the principal public buildings and organisations of a town, when the title begins with the town name.
Oxford University ( n o t the Oxford University)
Hull Station ( n o t the Hull Station)
Salisbury Cathedral
Manchester City Council
Birmingham Airport
Cheltenham Football Club
With the names of less important institutions, usage varies.
(The) East Oxford Community Centre. (The) Newbury School of English.
Names of single mountains vary. Most have no article.
Everest Kilimanjaro Snowdon Table Mountain
But definite articles are usually translated in the English versions of European mountain names, except those beginning Le Mont.
The Meije (= La Meije)
The Matterhorn (= Das Matterhorn)
b u t Mont Blanc ( n o t the Mont Blanc)
Did you mean to say "Mount Fuji" or "Fuji Mountain" in Japan? We generally don't use the article except in the case of ranges of mountains: the Andes, the Alps.