The kingdom of God is one of the names of Christ's church. This scripture is not referencing eternal heaven, but has reference to Christ's church. Baptism is an entrance into Christ's church, as a part of his bride. Other names the scriptures use to designate his church are; The church of Christ, the church in the wilderness, the new Jerusalem, the church of the first born, Zion, Israel (Jacob/Israel, not the nation of Israel), the remnant, the few, etc.
To be quickened, means to bring to life. We are first born a natural life, from our mother's womb. And at sometime in our life, the elect of God are quickened to a new spiritual life, with the indwelling of the Holy Spirit. This quickening event is explained in Eph 2.
Mankind is not quickened, because of them accepting spiritual things. At the time of their quickening, they were yet spiritually dead and unable to accept spiritual things, 1 Cor 2:14.