After attaining Nirvana will I remember that I was once without Nirvana? If yes , as Buddha did, then isn't it true that my relationship with or possession of Nirvana had a beginning?
I'm not sure how "I have attained", "I was once without", "my relationship with", and "my possession of" are compatible with Nibbana -- Nibbana is anatta, remember.
See also for example How do you know if you have attained Nibbana?
But yes the Buddha was able to remember and explain the details of his personal life (or lives), a sequence of events and experiences (including "the noble search") before enlightenment.
In Mahāyāna Abhidharma, we distinguish what is permanent (Skt. nitya) from what is eternal. That which is permanent is that which does not change from moment to moment. That which is eternal is that which is everlasting, never going out of existence.
Emptiness is permanent but occasional. This is because although emptiness is not produced and does not change from moment to moment as long as it abides (permanent), when the table ceases to exist, the emptiness of the table ceases to exist as well (occasional, not eternal).
With respect to nirvana, it is permanent, like all absences (the absence of inherent existence, the absence of John in the room, etc). But a Buddha has not always been established in nirvana. In this respect, it is occasional.
In addition, we usually say: although nirvana is not caused (because it's permanent), we can cause the achievement of nirvana. This is the whole point of the fourth noble truth, the path leading to the cessation of suffering.
When a monk's mind is thus freed, O monks, neither the gods with Indra, nor the gods with Brahma, nor the gods with the Lord of Creatures (Pajaapati), when searching will find on what the consciousness of one thus gone (tathaagata) is based. Why is that? One who has thus gone is no longer traceable here and now, so I say.
During a lifetime, nibbana is attained with remainder. The remainder is what supports that life until parinibbana. Memories are not a support, but nibbana is not annihalationism - there is no requirement to wipe any thing but further kamma, further intentional activity. So long as memories do not lead to further kamma, they are inconsequential.
Jumping in with the definition of Nibbana from the (Theravada) Abhidhamma
In the Abhidhamma there are four ultimate realities: consciousness, mental factors, matter, and Nibbana.
The first three (nama and rupa) are conditioned, thus anicca, dukkha and anatta.
Nibbana isn't; it is unarisen and unconditioned.
Nibbana is termed supramundane, and is to be realized by the knowledge of the four paths. It becomes an object to the paths and fruits, and is called Nibbana because it is a departure from craving, which is an entanglement.
Though Nibbana is onefold according to its intrinsic nature, by reference to a basis (for distinction), it is twofold, namely, the element of Nibbana with the residue remaining, and the element of Nibbana without the residue remaining. It is threefold according to its different aspects, namely, void, signless, and desireless.
An arahant experiences the element of Nibbana "with the residue remaining". The residue being the five aggregates. Though the defilements are eradicated, an arahant doesn't suddenly stop existing. The aggregates continue, meaning that sanna continues too. The body is still there, feelings are still there, memory and perception are still there.
Long story short, yes. An arahant will remember the times before experiencing the element of Nibbana.
Nibbāna causes four aggregates (nāma-khandha) by ārammaṇa-paccaya.
But no any aggregates (vaṭṭa-paṭiccasamuppāda) causes nibbāna (vivaṭṭa-paṭiccasamuppāda). So, nibbāna is not an effect of the dependent origination.
You, aggregates, can depend on nibbāna, but nibbāna never depend on you. So nibbāna is the opposite of paṭiccasamuppāda.
No one has a-nupādisesa-nibbāna in their hand, so no one own nibbāna. However the noble one can attain sa-upādisesa-nibbāna by attaining phala-samāpatti and saññāvedayita-nirodha-samāpatti.
In tipitaka and atthakathā, they often said "the noble one is owner of nibbāna", but it is just an idiom.