Tertullian uses the word "satisfaction" in several writings, but never, according to J. N. D. Kelly, to refer to Christ's atoning work on the cross. Kelly writes:

Tertullian has the theory that good deeds accumulate merit with God, while bad deeds demand 'satisfaction'—we observe the introduction of this important conception into Christian thought. Taken in conjunction with his doctrine of original sin, it might have enabled him to deal in a fresh way of his own with the problem of atonement. In fact, however, while using his ideas about satisfaction to explain the restoration of relations between the individual sinner and God, he altogether fails to apply them to the mediatorial role of Christ. (177)

Instead, Tertullian's uses of "satisfaction" come primarily in the contexts of repentance and good works, particularly baptism and martyrdom, which allow Christians to make satisfaction to God. This theme is most prominent in De Paenitentia ("On Repentance"). For example, people can make satisfaction to God through repentance or to the devil through apostasy:

Thus he who, through repentance for sins, had begun to make satisfaction to the Lord, will, through another repentance of his repentance, make satisfaction to the devil, and will be the more hateful to God in proportion as he will be the more acceptable to His rival. (5.9)

Along these lines, repentance is described as the "price" of pardon:

Further, how inconsistent is it to expect pardon of sins (to be granted) to a repentance which they have not fulfilled! This is to hold out your hand for merchandise, but not produce the price. For repentance is the price at which the Lord has determined to award pardon: He proposes the redemption of release from penalty at this compensating exchange of repentance. (6.4)

Again emphasizing that the Christian makes the satisfaction:

You will show your gratitude to the Lord by not refusing what the Lord offers you. You have offended, but can still be reconciled. You have One whom you may satisfy, and Him willing. (7.14; emphasis added)

In 8.9 he connects the more formal act of confession with satisfaction, and develops the idea more extensively in the following chapter, making it take the place of God's wrath:

This act, which is more usually expressed and commonly spoken of under a Greek name, is exomologesis, whereby we confess our sins to the Lord, not indeed as if He were ignorant of them, but inasmuch as by confession satisfaction is settled, of confession repentance is born; by repentance God is appeased. [...] All this exomologesis (does), that it may enhance repentance; may honour God by its fear of the (incurred) danger; may, by itself pronouncing against the sinner, stand in the stead of God's indignation, and by temporal mortification (I will not say frustrate, but) expunge eternal punishments. (9.2, 9.5)

A similar argument is made in De baptismo, 20, that confession of iniquties "make[s] satisfaction for our former sins." And in De Oratione, 7.1, he argues that God accepts penitence instead of "the death of the sinner."

Related themes, though not always the specific word "satisfaction," appear in other writings, such as Scorpiace, in which Tertullian argues that the sufferings of martyrs "pay" for spiritual benefits. In chapter 6, the martyr pays with his "flesh and life" in order to receive blessings from God. Martyrdom and the baptism of blood serve as "second supplies" of grace for those who sin following their conversions. He concludes:

For, strictly speaking, there cannot any longer be reckoned ought against the martyrs, by whom in the baptism (of blood) life itself is laid down. Thus, "love covers the multitude of sins;" and loving God, to wit, with all its strength (by which in the endurance of martyrdom it maintains the fight), with all its life (which it lays down for God), it makes of man a martyr. (6)

As Kelly notes, and Jaroslav Pelikan agrees, Tertullian does not connect the concept of satisfaction to Christ's death. But Kelly notes the possibility of a "germ" of substitution doctrine in Scorpiace (7), "It is Christ who gave Himself up for our offences," and Pelikan points to a passage in De Pudicitia, where the merit earned by martyrs is compared to that earned by Christ:

Let it suffice to the martyr to have purged his own sins: it is the part of ingratitude or of pride to lavish upon others also what one has obtained at a high price. Who has redeemed another's death by his own, but the Son of God alone? For even in His very passion He set the robber free. For to this end had He come, that, being Himself pure from sin, and in all respects holy, He might undergo death on behalf of sinners. Similarly, you who emulate Him in condoning sins, if you yourself have done no sin, plainly suffer in my stead. If, however, you are a sinner, how will the oil of your puny torch be able to suffice for you and for me? (22.4)

Conclusion

Thus we see that Ferguson's summary has merit. Tertullian does introduce the concept of satisfaction to soteriology, but he does not clearly extend it to refer to an objective work of Christ. G. D. Dunn argues that his focus was instead on the problem of sinful Christians, and quotes F. W. Dillistone:

Tertullian did not attempt to construct a doctrine of atonement in terms of his own particular interests for it was not an age for systematic theological exposition. Rather was it a period, especially in the West, of applying the Christian revelation to the urgent practical problems with which the Church had to deal. In Tertullian's time no problem was more poignant, even agonizing than that of how to deal with the weaker members of the fellowship who fell victim to open and grievous sins.

Sources and further reading:

• Dillistone, F. W., The Christian Understanding of Atonement, 187
• Dunn, G. D., "A Survey of Tertullian's Soteriology," Sacri. Erudiri 42 (2003): 61–86
• Kelly, J. N. D., Early Christian Doctrines, 177
• Pelikan, Jaroslav, The Christian Tradition: A History of Development of Doctrine, I, 147–48