It seems that the Catholic Answers website misattributed the quote to Pope Clement I.
The quote appears in Chapter 11 of the Epistle to Diognetus (AD 80-130), which can be read at the New Advent website: The Epistle of Mathetes to Diognetus.
The entirety of Chapter 11 with your quote highlighted:
I do not speak of things strange to me, nor do I aim at anything inconsistent with right reason; but having been a disciple of the Apostles, I have become a teacher of the Gentiles. I minister the things delivered to me to those that are disciples worthy of the truth. For who that is rightly taught and begotten by the loving Word, would not seek to learn accurately the things which have been clearly shown by the Word to His disciples, to whom the Word being manifested has revealed them, speaking plainly [to them], not understood indeed by the unbelieving, but conversing with the disciples, who, being esteemed faithful by Him, acquired a knowledge of the mysteries of the Father? For which reason He sent the Word, that He might be manifested to the world; and He, being despised by the people [of the Jews], was, when preached by the Apostles, believed on by the Gentiles. This is He who was from the beginning, who appeared as if new, and was found old, and yet who is ever born afresh in the hearts of the saints. This is He who, being from everlasting, is today called the Son; through whom the Church is enriched, and grace, widely spread, increases in the saints, furnishing understanding, revealing mysteries, announcing times, rejoicing over the faithful, giving to those that seek, by whom the limits of faith are not broken through, nor the boundaries set by the fathers passed over. Then the fear of the law is chanted, and the grace of the prophets is known, and the faith of the gospels is established, and the tradition of the Apostles is preserved, and the grace of the Church exults; which grace if you grieve not, you shall know those things which the Word teaches, by whom He wills, and when He pleases. For whatever things we are moved to utter by the will of the Word commanding us, we communicate to you with pains, and from a love of the things that have been revealed to us.
Other quotes concerning the Word of God in the Early Church Fathers period can be found at the Christian History for Everyman website.