According to the teachings of lord Buddha, what can we say about this quote? Is it fake?
"There is no path to happiness. Happiness is the path"
Apparently it wasn't the Buddha - it was A. J. Muste.
The American Dictionary of Quotations ascribes “There is no way to peace. Peace is the way” to a piece by Muste in the New York Times dated 16 November, 1967, although he definitely used the phrase well before that time.
The more relevant Buddha quote is perhaps the fourth noble truth
There is a way leading to the end of suffering, the eight-fold path, which is to be practiced.
The path to the end of suffering - which isn't quite the same as happiness to be fair.
When the path is actually entered, this is stream-entry. It is essentially a path of happiness since there is very little suffering, even at this entry level (SN 13.1).
The word 'vossagga' (relinquishment; letting go) is similar to (yet less mature than) the word 'paṭinissagga', which is included as a quality of Nibbana as defined in the 3rd noble truth. In the meditation teachings, it is said the beginning path levels of meditation have the quality of 'vossagga'.
Therefore, while the Buddha did not speak the quote in the OP, he did teach similar to it.
And what is the faculty of concentration? There is the case where a monk, a noble disciple, making letting go (vossagga) the meditation object, attains concentration, attains singleness of mind.
There is the case where a monk develops mindfulness as a factor for awakening dependent on seclusion, dependent on dispassion, dependent on cessation, resulting in relinquishment (vossagga).