The Bible does not say anything explicitly regarding this question. However, there are a number of things that can be deduced from what the Scriptures say.
The resurrection of the body
The first thing to keep in mind is that in Heaven, at the General Resurrection, we will have our bodies restored to us. This is found throughout the New Testament (and parts of the Old Testament as well), but probably the most explicit statement is from St. Paul in his first letter to the Corinthians:
So is it with the resurrection of the dead. What is sown is perishable; what is raised is imperishable. It is sown in dishonor; it is raised in glory. It is sown in weakness; it is raised in power. It is sown a natural body; it is raised a spiritual body. (1 Cor. 15:42-44, ESV; however, all of chapter 15 deals with this issue).
We could also mention Daniel 12:2, John 5:28-29, Acts 17:18, Matthew 22:29-32 and the parallel passages, as well as 2 Maccabees 7 (which for Catholics is considered canonical).
That means that, although (as St. Paul implies), our glorified bodies will have new and wonderful properties in Heaven, we will remain corporeal for all eternity.
Another observation to make is that, although God, in His Divine Nature, is pure spirit (as implied by passages such as Numbers 23:19), of course through the Incarnation, the Word, who is God, became flesh like us (John 1:14).
It follows that it will be possible to have physical interactions with Jesus (who is God), and presumably also private conversations. How exactly that will play out is difficult to imagine (e.g., there are an awful lot of us, and only one Jesus; however time will also work differently in Heaven), but since
He will wipe away every tear from their eyes (Rev. 21:4)
we can be sure that our desire to speak with Him will be well provided for.
Seeing God “face-to-face”
It should be noted, however, that our happiness in Heaven stems not only from being in the physical presence of Jesus (although that will be an important part of it, certainly), but above all from the so-called Beatific Vision. This Vision is described beautifully by St. Paul in 1 Corinthians:
For now we see in a mirror dimly, but then face to face. Now I know in part; then I shall know fully, even as I have been fully known (1 Cor. 13:12, which comes right after the hymn to charity).
Similarly, in the First Letter of St. John says,
Beloved, we are God’s children now, and what we will be has not yet appeared; but we know that when he appears we shall be like him, because we shall see him as he is (1 John 3:2).
This implies that we will, as it were, be immersed in the presence of God in Heaven. It will probably not be necessary, therefore, to have private conversations with Him, in order to obtain knowledge.
That does not mean that we will not enjoy conversations with Jesus, or what have you—just that we will have access to an even more perfect form of communication with God. Having the Vision is better than a private conversation, and it will certainly be available always, since it is the very essence of Heaven.
Prayer, as we know it on earth, is not needed in Heaven. We will be in direct contact with Him. And no, of course, there is no question of appointments. Time will exist in such a way that it will not be a limit on our access to God.
